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Algebra 1 Question on Rate, Distance, and Time

It is 74.4 miles from Dwane and Patti's house to Martha's house. Patti leaves home at 10:00 and drives toward Martha's house. Dwane and Martha set off half an hour later from their own houses and drive toward the other's house. The average speed of the three friends is 54 mph and they all meet at 11:06. How fast does each drive?

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Zakiyyah M. | Expert Math, Physics, and Engineering Tutor! Let's Succeed Together!Expert Math, Physics, and Engineering Tu...
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Sorry for my error. Please see Steve's answer.
 
Thanks for your correction Steve.  I assumed they traveled the same distance which was the flaw in my set up. Thanks again. 

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btw, from what i can tell, the distance from their houses is irrelevant information. 74.4 doesn't need to be used to solve the problem
The distance isn't irrelevant and your answers are wrong.
Please forgive me I haven't come up with another way.  I'll do more research and check again.  Do you have a suggested solution that I may review.  I probably should have worked the problem on my own before I uploaded my suggested process but as far as I can tell, this answer is okay.  I'll check again and update.
Are you assuming the distance each person traveled is the same?
Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
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*—————————————————*
D&P                   74.4 miles                    M

10:00 P starts
10:30 D & M start
11:06 all meet (at same point)

dD = dP is distance D & P traveled
dM + dD = 74.4 miles
(rD + rP + rM)/3 = 54 mph

11:06 - 10:00 = 1.1 hours = tP
11:06 - 10:30 = 0.6 hours = tD = tM

dD = rD*tD = 0.6 rD
dM = rM*tM = 0.6 rM
dP = rP*tP = 1.1 rP

0.6 rD = 1.1 rP; from dD = dP
0.6 rM + 0.6 rD = 74.4; from dM + dD = 74.4
(rD + rP + rM)/3 = 54; given

Eqn-1:    rD +     rP +   rM = 162
Eqn-2: 6 rD - 11 rP + 0 rM = 0
Eqn-3:    rD +  0 rP +   rM = 124

Eqn-1 - Eqn-3:

rP = 162 - 124 = 38 mph

Substitute in Eqn-2:

6 rD = 11(38)

rD = 11(2*19)/(2*3) = 11(19)/3 = 209/3 = 69 2/3 mph

Substitute in Equ-1:

209/3 + 38 + rM = 162

rM = 124 - (69 + 2/3) = 55 - 2/3 = 54 1/3 mph

Summary:

rD = 69 2/3 mph
rP = 38 mph
rM = 54 1/3 mph

check:

rD * tD = (69+2/3 mph)(0.6 hours) = 41.4 + 0.4 = 41.8 miles

rP * tP = (38 mph)(1.1 hours) = 41.8 miles = rD * tD √

rM * tM = (54 1/3 mph)(0.6 hours) = 32.4 + 0.2 = 32.6 miles

41.8 + 32.6 = 74.4 miles √

Comments

Thanks Steve.  That's exactly what I assumed. Distance was all the same.  I need to study your answer because I am having trouble understanding the concept for some reason. I keep thinking distance is irrelevant and I see that is incorrect.